# A second order analysis of McKean-Vlasov semigroups

## TL;DR

This paper develops a second order differential calculus for McKean-Vlasov semigroups, enabling detailed analysis of their regularity, stability, and propagation of chaos properties in nonlinear diffusion processes.

## Contribution

It introduces a novel second order calculus framework and explicit expansions for McKean-Vlasov semigroups, advancing understanding of their stability and chaos propagation.

## Key findings

- Provides second order Taylor expansions with remainders
- Derives Bismut-Elworthy-Li formulae for gradients and Hessians
- Establishes exponential decay inequalities for semigroup stability

## Abstract

We propose a second order differential calculus to analyze the regularity and the stability properties of the distribution semigroup associated with McKean-Vlasov diffusions. This methodology provides second order Taylor type expansions with remainder for both the evolution semigroup as well as the stochastic flow associated with this class of nonlinear diffusions. Bismut-Elworthy-Li formulae for the gradient and the Hessian of the integro-differential operators associated with these expansions are also presented. The article also provides explicit Dyson-Phillips expansions and a refined analysis of the norm of these integro-differential operators. Under some natural and easily verifiable regularity conditions we derive a series of exponential decays inequalities with respect to the time horizon. We illustrate the impact of these results with a second order extension of the Alekseev-Gr{\"o}bner lemma to nonlinear measure valued semigroups and interacting diffusion flows. This second order perturbation analysis provides direct proofs of several uniform propagation of chaos properties w.r.t. the time parameter, including bias, fluctuation error estimate as well as exponential concentration inequalities.